The term structure and the forward rates
Let $P(t,T)$ denote the price at time $t$ of a bond paying one unit of currency at time $T$. If we see $P(t,T)$ as a function of $T$, then this will represent the term structure of bond prices available at time $t$.
Let's suppose now that at time $t$ we adopt the following strategy: we buy one bond expiring at $T_2$, costing us $P(t,T_2)$. To finance this transaction, we sell $P(t,T_2)/P(t,T_1)$ units of a bond expiring at $T_1$. Since the inflow coming from the sale of this bond is equal to the outflow due to the purchase of the $T_2$ expiring bond, this strategy has a zero initial cost.
The net result of this strategy is that we'll have an outflow of $P(t,T_2)/P(t,T_1)$ at time $T_1$ and an inflow of $1$ at $T_2$. This means that this strategy allows us to lock an interest rate of
\begin{equation}\label{eq:fwdrate}
\frac{1}{\Delta}\left(\frac{P(t,T_1)}{P(t,T_2)}-1\right)
\end{equation}
between $T_1$ and $T_2$ (here $\Delta$ represents the time in years between these two points in time). The set of rates given by \ref{eq:fwdrate} are known as forward rates, and will be denoted as $F(t,T_1,T_2)$ which must be read as the forward rate from $T_1$ to $T_2$ as seen at time $t$.
Let's now introduce the concept of instantaneous forward rates. We can think of it as the forward rate when the tenor is very small, that is
\[
f(t,T)=\lim_{\Delta \to 0}F(t,T,T+\Delta)
\]
We can find a very useful expression for this quantity by expanding the right hand side
\[
f(t,T)=\lim_{\Delta \to 0}\frac{1}{\Delta}\left(\frac{P(t,T)}{P(t,T+\Delta)}-1\right)=
\]
\[
=\lim_{\Delta \to 0}\frac{1}{P(t,T+\Delta)}\left(\frac{P(t,T)-P(t,T+\Delta)}{\Delta}\right)=-\frac{1}{P(t,T)}\frac{\partial P(t,T)}{\partial T}=
\]
\[
=-\frac{\partial \text{ln}P(t,T)}{\partial T}
\]
